FM for Actuaries

276 CHAPTER 8
Example 8.13: Bond A is a 2-year annual coupon bond with coupon rate of
3%. Bond B is a 5-year annual coupon bond with coupon rate of 5.5%. You are
given that i S t =4.2%, 4.2%, 4.5%, 4.7% and 4.8%, for t =1, ··· , 5, respectively.
Compute the Fisher-Weil duration of the two bonds, as well as the price sensitivity
measure in (8.35). Also, calculate the Fisher-Weil duration of a portfolio with equal
weights in the two bonds.
Solution: For Bond A, we have C 1 =3and C 2 = 103. Using the given term
structure, we obtain PV(C 1 )=2.879 and PV(C 2 )=94.864, so that the price of
Bond A is 2.879 + 94.864 = 97.743. Consequently, we obtain W 1 =0.028 and
W 2 =1.863, so that the price sensitivity measure of Bond A as given in equation
(8.35) is 1.891 (i.e., Bond A drops in value by 1.891% per 1 percentage point
parallel increase in the term structure). Similarly, the Fisher-Weil duration of the
bond is computed as 1.971 years.
Similar calculations can be performed for Bond B, and the results are shown in
Table 8.3, with a Fisher-Weil duration of 4.510 years and a price sensitivity of 4.305
(i.e., Bond B drops in value by 4.305% per 1 percentage point parallel increase in
the term structure). To compute the Fisher-Weil duration of the portfolio, we take
the weighted average of the Fisher-Weil durations of the bonds to obtain 0.5(1.971
+ 4.510) = 3.241 years.
Table 8.6: Results of Example 8.13
A
B
t PV(C t ) W t tw t PV(C t ) W t tw t
1 2.879 0.028 0.029 5.278 0.049 0.051
2 94.864 1.863 1.941 5.066 0.094 0.098
3 4.820 0.134 0.140
4 4.577 0.169 0.177
5 83.454 3.858 4.044
Sum 97.743 1.891 1.971 103.194 4.305 4.510